Viscous Sublayer Research Articles

Data-Guided Low-Reynolds-Number Corrections for Two-Equation Models

Abstract The baseline Launder–Spalding k−ε model cannot be integrated to the wall. This paper seeks to incorporate the entire law of the wall into the model while preserving the original k−ε framework structure. Our approach involves modifying the unclosed dissipation terms in the k and ε equations specifically within the wall layer according to direct numerical simulation (DNS) data. The resulting model effectively captures the mean flow characteristics in both the buffer layer and the logarithmic layer, resulting in robust predictions of skin friction for zero-pressure-gradient (ZPG) flat-plate boundary layers and plane channels. To further validate our formulation, we apply our model to boundary layers under varying pressure gradients, channels experiencing sudden deceleration, and flow over periodic hills, with highly favorable results. Although not the focus of this study, the methodology here applies equally to the k–ω formulation and yields improved predictions of the mean flow in the viscous sublayer and buffer layer.

Heat transfer and fluid flow analysis of a solar air heater using conical jet impingement with sine-wave corrugation on absorber plate

In the pursuit of renewable energy solutions, solar air heaters emerge as a promising technology to efficiently harness solar thermal energy for diverse applications such as space heating, agricultural drying and industrial process heating. However, the system’s efficiency is often limited by the low heat transfer capability of air and the formation of laminar sub-layer over the absorber plate. This study addresses the issue by investigating the innovative use of conical jet impingement combined with sine wave corrugations on the absorber plate to enhance heat transfer in solar air heaters. The design parameters include wave amplitude ratio (0.079–0.314), throat jet diameter ratio (0.063–0.141), base jet diameter ratio (0.126–0.251), and Reynolds number (3200–19200) as the operational parameter. The computational fluid dynamics simulations are employed to solve the Reynolds-Averaged Navier-Stokes equations coupled with RNG k-ɛ turbulence model to assess the heat transfer and frictional characteristics, followed by experimental validation. The findings demonstrate a significant improvement in the system’s heat transfer rate with the Nusselt number increase by a factor of 6.71 times compared to a smooth solar air heater, while the friction factor increase by 13.87 times. Furthermore, the highest thermo-hydraulic performance parameter is found to be 2.79 corresponding to Reynolds number of 3200, wave amplitude ratio of 0.079, throat and base jet diameter ratios of 0.126 and 0.220, respectively. The proposed geometry will be helpful for academic researchers and can be adopted by the industries for various applications.

A quantitative three-dimensional inverse design rule based on cross-flow control of profiled end wall

Profiled end wall provides a novel and effective solution for end wall flow control in turbines and compressors. However, the application of profiled end walls in compressors still lacks quantitative design rules. This paper presents a quantitative design rule for end wall profiling based on a three-dimensional inverse method and numerically validates it on a highly loaded compressor cascade. The current inverse method can solve the corresponding end wall shape from a given end wall pressure distribution, but the determination of the end wall pressure distribution heavily relies on empirical knowledge. This study establishes a model between streamline curvature and cross-passage pressure gradient (CPG) through the circumferential equilibrium equation in the S1 stream surface, thereby providing a quantitative basis for determining the end wall pressure distribution. The quantitative design rule proposed in this paper is expressed as follows: at the axial position where the separation begins on the suction surface (SS), within the range of 0.1-0.2 pitch away from the SS, the end wall boundary layer fluid with a higher velocity than the corner region average velocity should possess the same streamline curvature as the fluid within the viscous sublayer. The inverse-designed profiled end wall using the quantitative design rule enhances the local cross-flow near the SS by imposing a stronger CPG, thus encouraging the end wall boundary layer fluid with relatively higher momentum to arrive at the SS earlier and enhancing the radial migration on the SS. Consequently, the intensified cross-flow entrains relatively higher momentum into the corner region, while the enhanced radial migration drives the low-momentum fluid away from the corner region towards the midspan. Finally, the inverse-designed profiled end wall reduces the half-span mass-flow-weighted average loss by 4.3%.

Improved Wall Function Method for Hypersonic Reacting Turbulent Boundary Layers

The chemical reaction and cold wall effects in hypersonic reacting turbulent boundary layers (HRTBLs) cause the failure of the existing wall function method. Therefore, this paper proposes an improved wall function method to simulate skin friction cf and wall heat flux qw in HRTBLs accurately and efficiently. Firstly, the temperature–velocity relation is extended to a generalized enthalpy–velocity relation to comprehensively involve the chemical reaction and cold wall effects. Then, a new unified velocity law of the wall for the inner layer of HRTBLs is theoretically derived. Finally, an improved wall function method is designed with the new laws of the wall. Numerical experiments of typical HRTBLs show that, compared with the existing wall function method, the improved wall function method can significantly reduce the errors of cf and qw from 8.0% and 10.2% to 0.4% and 0.5%, respectively, when the near-wall grids are relaxed to the viscous sublayer, while from 11.0% and 15.3% to 1.2% and 5.8%, respectively, when the near-wall grids are further relaxed to the logarithmic layer, bringing 2.7 and 15 times enhancement of the simulation efficiency. Moreover, by introducing the reference point, the improved wall function method removes the boundary layer edge quantities while ensuring accurate results.

Direct numerical simulation of 45° oblique flow past surface-mounted square cylinder

Comprehensive coherent structures around a surface-mounted low aspect ratio square cylinder in uniform flow with an oblique angle of $45^{\circ }$ were investigated for cylinder-width-based Reynolds numbers of 3000 and 10 000 by direct numerical simulation based on a topology-confined mesh refinement framework. High-resolution simulations and the critical-point concept were scrutinized to reveal for the first time the reasonable and compatible topologies of flow separation and complete near-wall structures, due to their extensive impact on various engineering applications. Large-scale horseshoe vortices are observed at two notable foci in the viscous sublayer. Within this layer, a wall-parallel jet is formed by downflow intruding into the bottom surface at the half-saddle point, then deflecting in the upstream direction and finally penetrating the bottom surface until the half-saddle point. A pair of conical vortices on the cylinder's top surface switch themselves on two sides of the square cylinder, where the switching frequency is identical with that of the sway of the side shear layer. The undulation of the Kelvin–Helmholtz instability is identified in the instantaneous development of a conical vortex and side shear layer, where the scaling of the ratio of the Kelvin–Helmholtz and von Kármán frequencies follows the power-law relation obtained by Lander et al. (J. Fluid Mech., vol. 849, 2018, pp. 1096–1119). Large-scale arch-shaped vortex is often detected in the intermediate wake region of a square cylinder, involving two interconnected portions, such as the leg portion separated from leeward surfaces and head portion rolled up from the top surface. The leg portion of the arch-shaped vortex was rooted by two foci near the bottom-surface plane.

Multi-cycle Direct Numerical Simulations of a Laboratory Scale Engine: Evolution of Boundary Layers and Wall Heat Flux

AbstractMulti-cycle direct numerical simulations (DNS) of a laboratory-scale engine at technically relevant engine speeds (1500 and 2500 rpm) are performed to investigate the transient velocity and thermal boundary layers (BL) as well as the wall heat flux during the compression stroke under motored operation. The time-varying wall-bounded flow is characterized by a large-scale tumble vortex, which generates vortical structures as the flow rolls off the cylinder wall. The bulk flow is found to strongly affect the development of the BL profiles, especially at higher engine speeds. As a result, the large-scale flow structures lead to alternating pressure gradients near the wall, invalidating the flow equilibrium assumptions used in typical wall modeling approaches. The thickness of the velocity BL and of the viscous sublayer was found to scale inversely with engine speed and crank angle. The thermal BL thickness also scales inversely with engine speed but increases with in-cylinder temperature. In contrast, thermal displacement thickness, which is sometimes used as a proxy for thermal BL thickness, was found to decrease with increasing temperature in the bulk. Examination of the heat flux distribution revealed areas of increased heat flux, particularly at places characterized by strong flow directed towards the wall. In addition, significant cyclic variations in the surface-averaged wall heat flux were observed for both engine speeds. An analysis of the cyclic tumble ratio revealed that the cycles with lower tumble ratio values near top dead center (TDC), indicative of an earlier tumble breakdown, also exhibit higher surface averaged wall heat fluxes. These findings extend previous numerical and experimental results for the evolution of BL structure during the compression stroke and serve as an important step for future engine simulations under realistic operating conditions.

Mean-flow structures of the turbulent boundary layers bounding a two-dimensional separation bubble

Understanding the mean-flow structures of a separated turbulent boundary layer (TBL) is crucial for turbulence modeling. This study investigates the spatial scaling properties of the total shear stress and mixing length in the TBLs bounding a two-dimensional (2D) separation bubble, aiming to derive analytical descriptions for the entire mean-velocity profiles of the TBLs. For the adverse pressure gradient (APG) TBL upstream of the separation bubble, the total shear stress possesses a two-layer structure with an inner layer adhering to a linear law and an outer layer conforming to a defect power law. In contrast, the mixing length profile consists of four layers, namely the viscous sublayer, the buffer layer, the overlap layer, and the wake region. Each of the layers exhibits a power law or a defect power law relationship with the spatial coordinate normal to the wall. In the four-layer structure, three parameters are sensitive to the variation of the APG: the buffer-layer thickness, the relative magnitude of the mixing length at the boundary layer edge, and a defect power law exponent quantifying the extent of the wake region. For the reattached TBL downstream of the separation bubble, the total shear stress consists of two parts. One part is induced by the pressure gradient and retains the two-layer structure, while the other, engendered by the intense turbulence advected from the separated shear layer, exhibits a dual-power-law distribution. The advected turbulence also significantly alters the four-layer structure of the mixing length, resulting in an augmented buffer layer, a diminished overlap layer, and a wake region that mimics a turbulent mixing layer. Via a dilation ansatz to describe the scaling transition between adjacent layers, the study formulates the complete profiles of the total shear stress and mixing length. The formulation leads to the derivation of novel analytical expressions for the entire mean-velocity profiles of the TBLs. The expressions are in precise accord with the direct numerical simulations of an incompressible 2D separation-bubble flow and a 2D impinging shock wave/TBL interaction. The elucidation of the mean-flow structures through this study is anticipated to facilitate the analysis of turbulence models, thereby enhancing their performance in simulating separated TBLs. The construction of the mean-flow descriptions by inspecting the spatial scaling properties of turbulence paves a promising way for theoretical exploration of complex nonequilibrium TBLs.

Turbulent boundary layer structures downstream of a square cylinder

Particle image velocimetry is used to investigate the coherent structures within a turbulent boundary layer (TBL) perturbed by a slender square cylinder. A moderate Reynolds number of Reτ=3691 is considered. The cylinder is immersed in different vertical regions of the TBL, from the viscous sublayer to the logarithmic layer, depending on the value of the cylinder–wall gap ratio within the range of G/D=0.0−2.0. Here, G is the distance between the wall and the lower surface of the cylinder, and D is the cross-section size of the square cylinder. The results reveal a substantial influence of the TBL statistics and structures. Utilizing velocity correlation and proper orthogonal decomposition (POD), we highlight the intriguing influence of large-/very large-scale turbulent structures induced by the cylinder. When the cylinder is positioned on or in close proximity to the wall, periodic shedding of vortices is not apparent. In the presence of a cylinder, the large-/very large-scale turbulent structures are reduced in size compared with those in the undisturbed TBL, owing to the interaction between the flow around the cylinder and the boundary layer turbulence. The lower modes of the POD in the disturbed TBL are similar to those of the undisturbed flow field. However, the large-/very large-scale turbulent structures are increasingly affected by the strengthening of the shedding vortices with increasing G/D when the cylinder is located in the logarithmic region. As a result, the Kármán vortex street becomes the dominant structure associated with the higher POD modes at sufficiently high cylinder–wall gap ratios.

Dynamic Wall Shear Stress Measurement Using Event-Based 3D Particle Tracking

We describe the implementation of a 3d Lagrangian particle tracking (LPT) system based on event-based vision (EBV) sensors and demonstrate its application for the near-wall characterization of a turbulent boundary layer (TBL) in air. The viscous sublayer of the TBL is illuminated by a thin light sheet that grazes the surface of a glass window inserted into the flat wall of the wind tunnel wall. The data simultaneously captured by three synchronized EBV-cameras is used to reconstruct the 3d particle tracks within 0.4 mm of the wall on a field of view of 12.0 mm by 7.5 mm. The velocity and position of particles within the viscous sub-layer permit the estimation of the local, unsteady wall shear stress vector under the assumption of linearity between particle velocity and wall shear stress. Thereby, 2d distributions and higher order statistics of the unsteady wall shear stress are obtained. The systematic underestimation of the spanwise shear stress fluctuation can be explained on the basis of DNS data; a methodology for its mitigation is proposed. The employed EBV hardware coupled with suited LPT algorithms provide data quality on par with currently used, considerably more expensive, high-speed framing cameras.

Wall Shear Stress Characteristics Of A Turbulent Boundary Layer Obtained With Multi-Aperture Defocusing Micro-PTV And High-Speed Stereo Profile-PIV

A multi-aperture 3d 3c micro PTV system is presented which relies on single window optical access for both high-speed tracer illumination and image recording. Similar to the “defocusing” concept described by C. Willert & Gharib (1992), the wall distance of individual particles is obtained from the size of projected particle image triplets formed by a triplet of apertures on the entrance pupil of the microscope lens. The present article extends upon previously published material Klinner & Willert (2022) by applying multi-aperture micro particle tracking velocimetry (MA-micro PTV) on a canonical turbulent boundary layer (TBL) to track the near-wall motion of tracer particles with the aim of estimating the unsteady wall-shear stress from particle tracks. The technique is validated with measurements of a TBL inside the closed test section of the 1 m wind tunnel of DLR in Göttingen (1mWK) at free- stream velocities of 5.2 to 20 m/s with corresponding shear Reynolds numbers between 560 and 1630. In the viscous sublayer, the probability density distributions of stream- and span-wise wall shear stress (WSS) components could be reliably captured down to probability densities of 10E−3. As far as we know, this has not been achieved in the past, particularly not for the span-wise component. For the stream-wise component the measured Reynolds-number dependency of the root mean squared fluctuations agrees to correlations by Örlü & Schlatter (2011). Furthermore, the skewness of the stream-wise WSS agrees well to values from DNS. On the other hand, the span-wise WSS fluctuations consistently underestimate the predicted DNS values, for which it is assumed that the deviation can be explained by the rapid decrease of the span-wise velocity fluctuations with increasing wall distance. Wall-normal profiles of 3c velocity statistics were obtained by bin averaging of particle velocities. Up into the buffer layer, these profiles agree well with profiles from stereo PIV as well as from DNS and LES, indicating an accurate determination of both near-wall velocities and inner scaling.

Rainbow Imaging Of 3-D Microbubble Distribution Clustered Inside A Turbulent Boundary Layer

Microbubbles dispersed in a turbulent boundary layer spatially developing along a flat plate is investigated experimentally. The aim of study is to find out visually how microbubbles interact with turbulent eddies to achieve frictional drag reduction on a wall. For stream Reynolds number up to 3×104 , microbubble suspended at Stokes number smaller than 10 − 3 is measured. Stream-wise vortices in the boundary layer were visualized by flake optics, of which spacing expands with injection of microbubbles. 3-D microbubble distributions were measured by a color-coded volumetric illumination applied to the boundary layer. The result showed preferentially accumulated microbubble clouds to low-speed streaks close to viscous sublayer and to hair-pin vortices in the buffer layer.

Near-Wall Flow Features In ZPG-TBL At Various Reynolds Numbers Using Dense 3D Lagrangian Particle Tracking

The 3D Lagrangian particle tracking method Shake-The-Box (STB) and, subsequently, the data assimilation method FlowFit3 have been applied to measure the flow field in a narrow wall-parallel volume inside the nearwall region of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) flow at various Reynolds numbers. The STB experiments have been conducted in the 1m- wind tunnel facility of DLR in Göttingen at relatively high particle image densities (~0.07 ppp) using µm-sized DEHS particles and high-repetition pulse laser at up to 38.1 kHz. The results show that the chosen methodology is able to volumetrically resolve almost all features of the flow inside the viscous sub-, buffer- and lower logarithmic- layer in space and time at Reynolds numbers up to Re θ = 5,455. The gained data consists of time-series of spatially well resolved 3D velocity, acceleration and pressure fields, as well as a huge amount of long Lagrangian particle trajectories. Selecting the viscous sublayer part of the measurement volume provides time-series of the 2D distribution of both components of the instantaneous wall-shear stresses w,x/z . The resulting data are used for conditional and time-resolved 3D analyses of coherent structures, rare (e.g. back flow) events and two-point space-timecorrelations.

Drag Reduction by Dried Malted Rice Solutions in Pipe Flow

In this study, the friction factor of a turbulent pipe flow for dried rice malt extract solutions was experimentally reduced to that of a Newtonian fluid. The friction factor was measured for four types of solutions at different culture times and concentrations. The results indicate that the experimental data points of the test solutions diverged from the maximum drag reduction asymptote at and above Re√f ≅ 200~250 and aligned parallel to those of Newtonian fluids. This drag reduction phenomenon differed from that observed in artificial high-molecular-weight polymer solutions, called Type A drag reduction, in which the drag reduction level is dependent on the Reynolds number in the intermediate region. This is classified as a Type B drag reduction phenomenon in biopolymer solutions and fine solid particle suspensions. The order of drag reduction corresponded to approximately 5–50 ppm xanthan gum solutions, as reported previously. Furthermore, the velocity profile in a turbulent pipe flow was predicted using a semi-theoretical equation in which the friction factors were determined using the difference between the experimental results of the tested solutions and Newtonian fluids. The results indicate considerable thickening of the viscous sublayer in the turbulent pipe flow of the test solutions compared with that of Newtonian fluids.

Conjugate heat transfer analysis of helium–xenon gas mixture in tight rod bundles

Helium-xenon gas mixture (He-Xe) and tight rod bundles are widely used in gas-cooled space nuclear reactors. However, few studies have been conducted on the effects of pitch-to-diameter ratio (P/D) on thermal–hydraulic characteristics while considering conjugate heat transfer. This paper numerically investigated He-Xe heat transfer in a fuel-gas gap-cladding-coolant structure of tight rod bundles. The numerical model is validated using experimental data and the maximum error of wall temperature is less than 5 %. The results show that conjugate heat transfer significantly affects the heat transfer. The heat transfer coefficient without conjugate heat transfer is only 8.85 % of that with conjugate heat transfer in the quasi-triangular pipe. The decrease in P/D enhances the convective heat transfer and causes larger pressure drops. A new performance evaluation criterion (PEC) is presented considering the effects of heat transfer, pressure drop and volume. PEC first increases and then decreases with the increase in P/D and the best comprehensive performance occurs at P/D = 1.113. The analysis of the temperature difference in the viscous sublayer (y+ < 5) shows that the decrease in P/D enhances the turbulent effect of He-Xe and reduces the thickness of the viscous sublayer, resulting in the heat transfer enhancement. This study contributes to the optimal design of advanced space reactors.

Mechanism analysis on heat transfer of supercritical LNG in horizontal U-bend tube

A comprehensive comprehension of the intricate heat transfer mechanisms pertaining to supercritical fluids within U-bend tubes holds paramount significance in the optimization design of submerged combustion vaporizers and other water-bath heating heat exchangers. This paper employs a combined approach of experimental analysis and numerical simulation to explore the complicated heat transfer processes of supercritical liquefied natural gas in the U-bend tube. For a careful consideration of similarity and safety, the N2–Ar mixture is used as the alternative experimental media. Upon validating the model's accuracy using experimental data, this study unveils the mechanism of the coupling effect between centrifugal force and buoyancy on heat transfer based on simulation results. The results show that the internal circulation flow field structure caused by centrifugal force weakens the heat transfer on the inner side within an angle range of approximately 60°. Along the entrance region of downstream straight tubing, secondary flow continues to be predominantly governed by centrifugal forces due to inertia-driven effects. Subsequently, buoyancy-driven secondary flow impels compression upon centrifugally-induced secondary flow towards the inner side, thereby instigating significant augmentation in fluidic heat transfer resistance in both the viscous sub-layer and buffer layer of the inner side. Overall, the downstream straight tube section exhibits a special heat transfer variation pattern characterized by an initial strengthening, followed by deterioration, and ultimately recovery due to the influence of centrifugal force, with a maximum influence distance extending to 175.5d.

Theoretical evaluation of a hybrid buoyancy-compressed air energy storage system

Energy storage plays a pivotal role in the emerging green economy. This study, for the first time, presents the theoretical evaluation of a buoyancy power generator combining with the compressed air energy storage (CAES-BPG) system. A theoretical model that satisfies conservation laws and does not involve any adjustable parameters is developed for the calculation of system efficiency and power generation capacity. To accurately predict drag acting on the buoy, the low Reynolds number κ-ω SST model with a locally refined mesh (y+~1) is applied to fully resolve the viscosity affected regions including the viscous sublayers, whilst the surrounding flow is modelled by LES. The refined mesh moves passively with the buoy and does not change, whilst mesh in other regions undergoes smoothing and remeshing to accommodate the buoy's motion. Correlations of drag coefficients are then developed as a function of buoy shape, Reynolds number, and vessel-to-buoy size ratio. The aerofoil shape yielded a substantially smaller drag coefficient compared to the square, hex, and bullet shapes. System efficiency showed independency on buoy volume and shell material but decreased as the water level in the vessel increased, reaching 64 % for a water level of 1.5 m. Similarly, as the water level increased, the system output power decreased. The output power increased as the buoy volume increased, reaching ~22 kW for a buoy volume of 2 m3 at a water level of 1.5 m. This CAES-BPG system is highly automatic and fully green, and can operate continuously once compressed air is connected, showing great potential for implementation as an energy storage / utilisation system of small to medium capacities.

Erosion rate of lunar soil under a landing rocket, part 1: Identifying the rate-limiting physics

Multiple nations are planning activity on the Moon's surface, and to deconflict lunar operations we must understand the sandblasting damage from rocket exhaust blowing soil. Prior research disagreed over the scaling of the erosion rate, which determines the magnitude of the damage. Reduced gravity experiments and two other lines of evidence now indicate that the erosion rate scales with the kinetic energy flux at the bottom of the laminar sublayer of the gas. Because the rocket exhaust is so fast, eroded particles lifted higher in the boundary layer do not impact the surface for kilometers (if at all; some leave the Moon entirely), so there is no saltation in the vicinity of the gas. As a result, there is little transport of gas kinetic energy from higher in the boundary layer down to the surface, so the emission of soil into the gas is a surprisingly low energy process. In low lunar gravity, a dominant source of resistance to this small energy flux turns out to be the cohesive energy density of the lunar soil, which arises primarily from particles in the 0.3 to 3 μm size range. These particles constitute only a tiny fraction of the mass of lunar soil and have been largely ignored in most studies, so they are poorly characterized.

Numerical prediction of mean flow and acoustic field of a supersonic impinging jet

The three-dimensional turbulent mean flow and acoustic field of a supersonic jet impinging on a solid plate is studied computationally using the general purpose CFD code Ansys Fluent. A pressure-based coupled solver formulation with the second order weighted central-upwind spatial discretization is applied to compute transient solutions. Cold and hot jet thermal conditions are considered. Mean flow characteristics are investigated by a steady-state modeling approach. Acoustic radiation of impingement tones is simulated using a transient time-domain formulation. The effects of turbulence in steady-state are modeled by the SST k-ω turbulence model. The Wall-Modeled Large-Eddy Simulation (WMLES) model is applied to compute transient solutions. The near-wall mesh on the impingement plate is fine enough to resolve the viscosity-affected near-wall region all the way to the laminar sublayer. Nozzle-to-plate distance is parameterized in the model for automatic re-generation of the mesh and results. Steady-state predictions of hover lift loss and mean jet velocity distributions are compared with experimental data, and favorable agreement is reported. The transient solution reproduces the mechanism of impingement tone generation by the interaction of large scale vortical structures with the impingement plate. The acoustic near-field is directly resolved by Computational Aeroacoustics (CAA) to accurately propagate impingement tone waves to near-field microphone locations. Calculated impingement tone frequencies and sound pressure levels agree with experimental values.

Turbulent drag reduction using dolphin-inspired near-wall ultrasonic microvibrations

The skin-friction drag generated by wall-bounded turbulent flows can potentially be reduced by a wall-parallel oscillatory motion. Inspired by microvibrations and the high sensitivity of dolphin skin, we examine whether wall-normal undulating motion actuated by longitudinal micro-ultrasonic waves (LMUWs) with ultrasonic-frequency oscillations and micro-size amplitudes significantly alters the multi-eddy motion on the surface, thereby reducing skin-friction drag. Simulations of the LMUW-induced turbulent flows are performed in an open channel at a Reynolds number of 1.24 × 106 for three motion modes, i.e., two traveling waves (downstream and upstream) in the streamwise direction and a standing wave. It is verified that the wall-normal turbulent fluctuations are remarkedly altered within the viscous sublayer of the turbulent boundary layer, resulting in a reduced velocity gradient. This leads to lower or even extinguished friction drag, which is strongly associated with the LMUW-excitation mode. Informed and validated by numerical results, we further derived a theoretical model for the dynamic boundary layer. This model is based on Fourier series expressions of the velocities and is used to elucidate the underlying mechanisms in association with the LMUW-excited turbulent flow and active friction drag reduction. The results indicate that upstream traveling waves enable 100% friction drag reduction, while downstream traveling waves are capable of overcoming the trade-off between friction and pressure drag, accomplishing 100% total drag reduction. This study thus provides a novel active and controllable method for turbulent drag reduction.

Experimental assessment of square-wave spatial spanwise forcing of a turbulent boundary layer

We present an experimental realisation of spatial spanwise forcing in a turbulent boundary layer flow, aimed at reducing the frictional drag. The forcing is achieved by a series of spanwise running belts, running in alternating spanwise direction, thereby generating a steady spatial square-wave forcing. Stereoscopic particle image velocimetry in the streamwise–wall-normal plane is used to investigate the impact of actuation on the flow in terms of turbulence statistics, drag performance characteristics, and spanwise velocity profiles, for a non-dimensional wavelength of λx+=397\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _x^+ = 397$$\\end{document}. In line with reported numerical studies, we confirm that a significant flow control effect can be realised with this type of forcing. The scalar fields of the higher-order turbulence statistics show a strong attenuation of stresses and production of turbulence kinetic energy over the first belt already, followed by a more gradual decrease to a steady-state energy response over the second belt. The streamwise velocity in the near-wall region is reduced, indicative of a drag-reduced flow state. The profiles of the higher-order turbulence statistics are attenuated up to a wall-normal height of y+≈100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y^+ \\approx 100$$\\end{document}, with a maximum streamwise stress reduction of 45% and a reduction of integral turbulence kinetic energy production of 39%, for a non-dimensional actuation amplitude of A+=12.7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+ = 12.7$$\\end{document}. An extension of the classical laminar Stokes layer theory is introduced, based on the linear superposition of Fourier modes, to describe the non-sinusoidal boundary condition that corresponds to the current case. The experimentally obtained spanwise velocity profiles show good agreement with this extended theoretical model. The drag reduction was estimated from a linear fit in the viscous sublayer in the range 2≤y+≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \\le y^+\\le 5$$\\end{document}. The results are found to be in good qualitative agreement with the numerical implementations of Viotti et al. (Phys Fluids 21, 2009), matching the drag reduction trend with A+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+$$\\end{document}, and reaching a maximum of 20%.Graphical abstract